Question: All of the 5th grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$4.50$ each for students, and the group paid $$43.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$14.00$ each for teachers and $$12.50$ each for students, and the group paid $$104.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+4.5y = 43.5}$ ${14x+12.5y = 104.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-14x-9y = -87}$ ${14x+12.5y = 104.5}$ Add the top and bottom equations together. $ 3.5y = 17.5 $ $ y = \dfrac{17.5}{3.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {7x+4.5y = 43.5}$ to find $x$ ${7x + 4.5}{(5)}{= 43.5}$ $7x+22.5 = 43.5$ $7x = 21$ $x = \dfrac{21}{7}$ ${x = 3}$ You can also plug ${y = 5}$ into $ {14x+12.5y = 104.5}$ and get the same answer for $x$ ${14x + 12.5}{(5)}{= 104.5}$ ${x = 3}$ There were $3$ teachers and $5$ students on the field trips.